Quantitative Darboux theorems in contact geometry
Authors:
John B. Etnyre, Rafal Komendarczyk and Patrick Massot
Journal:
Trans. Amer. Math. Soc. 368 (2016), 7845-7881
MSC (2010):
Primary 53D10, 53D35; Secondary 57R17
DOI:
https://doi.org/10.1090/tran/6821
Published electronically:
March 1, 2016
MathSciNet review:
3546786
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Abstract | References | Similar Articles | Additional Information
Abstract: This paper begins the study of relations between Riemannian geometry and contact topology on $(2n+1)$–manifolds and continues this study on 3–manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact $(2n+1)$–manifold $(M,\xi )$ that can be embedded in the standard contact structure on $\mathbb {R}^{2n+1}$, that is, on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form $\alpha$ for $\xi$. In dimension 3, this further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curve techniques to provide a lower bound for the radius of a PS-tight ball.
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Additional Information
John B. Etnyre
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
MR Author ID:
619395
Email:
etnyre@math.gatech.edu
Rafal Komendarczyk
Affiliation:
Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email:
rako@tulane.edu
Patrick Massot
Affiliation:
Département de Mathématiques, Université Paris Sud, 91405 Orsay Cedex, France
Address at time of publication:
Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France
MR Author ID:
844086
Email:
patrick.massot@polytechnique.edu
Received by editor(s):
September 17, 2012
Received by editor(s) in revised form:
January 9, 2015
Published electronically:
March 1, 2016
Article copyright:
© Copyright 2016
American Mathematical Society